Question

(A) If \(\frac{1}{x}:\frac{1}{y}:\frac{1}{z}\) =2: 3: 5, then x:y:z = 15:10: 6
(B) If 4p = 6q = 9r then p:q:r = :9:6:4
(C) If 2A = 3B = 4C then A:B:C = 3:4:6
(D) P:Q:R = 2:3:4 then \(\frac{P}{Q}:\frac{Q}{R}:\frac{R}{P}\)=9:8:24
Choose the correct answer from the options given below:

• (C) only
• (D) only
• (A) and (B) only
• (C) and (D) only

Answer 1

The Correct Option is C

1. Let's first evaluate statement (A): If \(\frac{1}{x}:\frac{1}{y}:\frac{1}{z} = 2:3:5\), then we can find x:y:z.
   • Given \(\frac{1}{x}:\frac{1}{y}:\frac{1}{z} = 2:3:5\), we can write this as:
     • \(\frac{1}{x} = 2k, \frac{1}{y} = 3k, \frac{1}{z} = 5k\) for some constant \(k\).
   • This implies:
     • \(x = \frac{1}{2k}\)
     • \(y = \frac{1}{3k}\)
     • \(z = \frac{1}{5k}\)
   • Thus, the ratio \(x:y:z\) is:
     • \(x : y : z = \frac{1}{2k} : \frac{1}{3k} : \frac{1}{5k} = \frac{1}{2} : \frac{1}{3} : \frac{1}{5}\)
     • By taking the reciprocal, \(x : y : z = \frac{1}{1/2} : \frac{1}{1/3} : \frac{1}{1/5}\)
     • This simplifies to \(x : y : z = 15 : 10 : 6\).
   • Statement (A) is correct.
2. Next, evaluate statement (B): If \(4p = 6q = 9r\), then we can find \(p:q:r\).
   • Given \(4p = 6q = 9r = k\) for some constant \(k\).
   • We have:
     • \(p = \frac{k}{4}\)
     • \(q = \frac{k}{6}\)
     • \(r = \frac{k}{9}\)
   • The ratio \(p:q:r\) is:
     • \(p:q:r = \frac{k}{4} : \frac{k}{6} : \frac{k}{9} = 9 : 6 : 4\) after simplifying each fraction to a common multiple.
   • Statement (B) is correct.
3. Evaluate statement (C): If \(2A = 3B = 4C\), find \(A:B:C\).
   • For \(2A = 3B = 4C = k\), we have:
     • \(A = \frac{k}{2}\)
     • \(B = \frac{k}{3}\)
     • \(C = \frac{k}{4}\)
   • The ratio \(A : B : C\) is:
     • \(A : B : C = \frac{k}{2} : \frac{k}{3} : \frac{k}{4} = 6 : 4 : 3\), but the statement claims \(3 : 4 : 6\). This is incorrect.
4. Evaluate statement (D): If \(P:Q:R = 2:3:4\), then calculate \(\frac{P}{Q}:\frac{Q}{R}:\frac{R}{P}\).
   • Given \(P:Q:R = 2:3:4\), let's find the given ratio:
     • \(\frac{P}{Q} = \frac{2}{3}\)
     • \(\frac{Q}{R} = \frac{3}{4}\)
     • \(\frac{R}{P} = \frac{4}{2} = 2\)
   • The ratio \(\frac{P}{Q}:\frac{Q}{R}:\frac{R}{P}\) becomes:
     • \(\frac{2}{3} : \frac{3}{4} : 2 = 8 : 9 : 24\)
     • This is different from the given \(9 : 8 : 24\).
   • Statement (D) is incorrect.

Therefore, the correct answer is (A) and (B) only.